Problem: Bela started studying how the number of branches on her tree grows over time. Every $2.9$ years, the number of branches increases by an additional $83\%$, and can be modeled by a function, $N$, which depends on the amount of time, $t$ (in years). When Bela began the study, her tree had $60$ branches. Write a function that models the number of branches $t$ years since Bela began studying her tree. $N(t) = $
Answer: The strategy We can model the situation with an exponential function of the general form A ⋅ B f ( t ) A\cdot B\^{ f(t)}, where $A$ is the initial quantity, $B$ is a factor by which the quantity is multiplied over constant time intervals, and $f(t)$ is an expression in terms of $t$ that determines those time intervals. Let's use the given information to determine $A$, $B$, and $f(t)$. Understanding what's given We are given that the initial number of branches is $60$, and every $2.9$ years, the number of branches increases by an additional $83\%$. Note that increasing an additional $83\%$ is the same as being multiplied by $1.83$. [Why?] This means that the initial quantity is $A=60$ and the factor is $B=1.83$. We need to find $f(t)$ based on the fact that the quantity is multiplied by $1.83$ every $2.9$ years. Finding the expression in the exponent We know that the number of branches is multiplied by $1.83$ every $2.9$ years. This means that each time $t$ increases by $2.9$, $f(t)$ increases by $1$. Therefore, $f(t)$ is a linear function whose slope is $\dfrac{1}{2.9}$. When the initial measurement is made, the number of branches hasn't changed. So $N(0) = 60$, which means that $f(0)=0$. [Why?] Therefore, $f(t)$ must be $\dfrac{t}{2.9}$. Summary We found that the following function models the number of branches $t$ years since Bela began studying her tree. N ( t ) = 60 ⋅ ( 1.83 ) t 2.9 N(t)=60\cdot (1.83)\^{ \frac{t}{2.9}}